Practice Questions

Q87.For a, b ∈Z and |a −b| ≤10 , let the angle between the plane P : a x + y −z = b and the line L : x −1 = a −y = z + 1 be cos−1( 13 ) If the distance of the point (6, −6, 4) from the plane P is 3√6 , then a4 + b2 is equal to (1) 32 (2) 85 (3) 25 (4) 48

Q87.Let the plane containing the line of intersection of the planes P1 : x + (λ + 4)y + z = 1 and P2 : 2x + y + z = 2 pass through the points (0, 1, 0) and (1, 0, 1) . Then the distance of the point (2λ, λ, −λ) from the plane P2 is (1) 5√6 (2) 4√6 (3) 2√6 (4) 3√6

Q87.Let the plane P : 8x + α1y + α2z + 12 = 0 be parallel to the line L : x+22 = y−33 = z+45 . If the intercept of P on the y-axis is 1 , then the distance between P and L is (1) √27 (2) √146 (3) √72 (4) √14 JEE Main 2023 (31 Jan Shift 2) JEE Main Previous Year Paper

Q87.The distance of the point P(4, 6, −2) from the line passing through the point (−3, 2, 3) and parallel to a line with direction ratios 3, 3, −1 is equal to: (1) 3 (2) √6 (3) 2√3 (4) √14

Q88.Let the line passing through the points P(2, −1, 2) and Q(5, 3, 4) meet the plane x −y + z = 4 at the point R. Then the distance of the point R from the plane x + 2y + 3z + 2 = 0 measured parallel to the line x−7 2 = y+32 = z−21 is (1) √61 (2) √189 (3) √31 (4) 3

Q88.If a plane passes through the points (−1, k, 0), (2, k, −1), (1, 1, 2) and is parallel to the line x−11 = 2y+12 = z+1−1 , then the value of (k−1)(k−2)k2+1 is (1) 17 (2) 5 5 17 (3) 6 (4) 13 13 6

Q88.If the foot of the perpendicular drawn from (1, 9, 7) to the line passing through the point (3, 2, 1) and parallel the planes x + 2y + z = 0 and 3y −z = 3 is (α, β, γ), then α + β + γ is equal to (1) −1 (2) 3 (3) 1 (4) 5 y−√6

Q88.Let the plane P : 4x −y + z = 10 be rotated by an angle π2 about its line of intersection with the plane x + y −z = 4 . If α is the distance of the point (2, 3, −4) from the new position of the plane P , then 35α is equal to (1) 85 (2) 105 (3) 126 (4) 90

Q88.Let P be the plane passing through the line x−1 1 = y−2−3 = z+57 and the point (2, 4, −3). If the image of the point (−1, 3, 4) in the plane P is (α, β, γ), then α + β + γ is equal to (1) 10 (2) 12 (3) 9 (4) 11

Q88.If the equation of the plane containing the line x + 2y + 3z −4 = 0 = 2x + y −z + 5 and perpendicular to + + + ax + by + cz = 4 then (a −b + c) is equal to the plane→r= (ˆi −ˆj) λ(ˆi + ˆj + ˆk) μ(ˆi −2ˆj 3ˆk) is (1) 18 (2) 22 (3) 20 (4) 24

Q88.Consider the lines L1 and L2 given by L1 : x−12 = y−31 = z−22 L2 : x−21 = y−22 = z−33 A line L3 having direction ratios 1, −1, −2, intersects L1 and L2 at the points P and Q respectively. Then the length of line segment PQ is (1) 2√6 (2) 3√2 (3) 4√3 (4) 4

Q88.The distance of the point (−1, 2, 3) from the plane →r⋅(ˆi −2ˆj + 3ˆk) is + + + + distance between the lines→r= (ˆi −ˆj) λ(2ˆi ˆk) and →r= (2ˆi −ˆj) μ(ˆi −ˆj ˆk) (1) 4√6 (2) 2√5 (3) 2√6 (4) 3√5

Q88.Let P be the plane, passing through the point (1, −1, −5) and perpendicular to the line joining the points (4, 1, −3) and (2, 4, 3). Then the distance of P from the point (3, −2, 2) is (1) 6 (2) 4 (3) 5 (4) 7

Q88.A plane P contains the line of intersection of the plane →r⋅(ˆi + ˆj + ˆk) = 6 and →r⋅(2ˆi + 3ˆj + 4ˆk) passes through the point (0, 2, −2),then the square of distance of the point (12, 12, 18) from the plane P is (1) 620 (2) 155 (3) 310 (4) 1240

Q88.The plane 2x −y + z = 4 intersects the line segment joining the points A(a, −2, 4) and B(2, b, −3) at the point C in the ratio 2 : 1 and the distance of the point C from the origin is √5 . If ab < 0 and P is the point (a −b, b, 2 b −a) then CP2 is equal to: (1) 17 (2) 16 3 3 (3) 73 (4) 97 3 3

Q89.Fifteen football players of a club-team are given 15 T-shirts with their names written on the backside. If the players pick up the T-shirts randomly, then the probability that at least 3 players pick the correct T-shirt is (1) 5 (2) 2 24 15 (3) 1 (4) 5 6 36

Q89.Two dice A and B are rolled. Let the numbers obtained on A and B be α and β respectively. If the variance of α −β is pq , where p and q are co-prime, then the sum of the positive divisors of p is equal to (1) 72 (2) 36 (3) 48 (4) 31

Q89.The foot of perpendicular from the origin O to a plane P which meets the co-ordinate axes at the point A , B, C is (2, a, 4), a ∈N . If the volume of the tetrahedron OABC is 144 unit3 , then which of the following points is NOT on P ? (1) (0, 4, 4) (2) (3, 0, 4) (3) (0, 6, 3) (4) (2, 2, 4)

Q89.If the lines x−1 1 = y−22 = z+31 and x−a2 = y+23 = z−31 intersects at the point P , then the distance of the point P from the plane z = a is : (1) 16 (2) 28 (3) 10 (4) 22 n ≥2

Q90.Two dice are thrown independently. Let A be the event that the number appeared on the 1st die is less than the number appeared on the 2nd die, B be the event that the number appeared on the 1st die is even and that on the second die is odd, and C be the event that the number appeared on the 1st die is odd and that on the 2nd is even. Then JEE Main 2023 (01 Feb Shift 2) JEE Main Previous Year Paper (1) The number of favourable cases of the event (2) A and B are mutually exclusive (A ∪B) ∩C is 6 (3) The number of favourable cases of the events A , (4) B and C are independent B and C are 15, 6 and 6 respectively JEE Main 2023 (01 Feb Shift 2) JEE Main Previous Year Paper

Q90.A coin is biased so that the head is 3 times as likely to occur as tail. This coin is tossed until a head or three tails occur. If X denotes the number of tosses of the coin, then the mean of X is (1) 38 (2) 15 16 16 (3) 21 (4) 81 16 64 JEE Main 2023 (13 Apr Shift 1) JEE Main Previous Year Paper

Q90.Let S = {w1, w2, … . } be the sample space associated to a random experiment. Let P(wn) = P(wn−1)2 , . Let A = {2k + 3l; k, l ∈N} and B = {wn; n ∈A} . Then P(B) is equal to (1) 3 (2) 3 32 64 (3) 1 (4) 1 16 32 JEE Main 2023 (29 Jan Shift 2) JEE Main Previous Year Paper

Q90.In a bolt factory, machines A, B and C manufacture respectively 20%, 30% and 50% of the total bolts. Of their output 3, 4 and 2 percent are respectively defective bolts. A bolt is drawn at random from the product. If the bolt drawn is found the defective then the probability that it is manufactured by the machine C is (1) 5 (2) 9 14 28 (3) 3 (4) 2 7 7 JEE Main 2023 (08 Apr Shift 1) JEE Main Previous Year Paper

Q90.There rotten apples are mixed accidently with seven good apples and four apples are drawn one by one without replacement. Let the random variable X denote the number of rotten apples. If μ and σ2 represent mean and variance of X, respectively, then 10(μ2 + σ2) is equal to (1) 20 (2) 250 (3) 25 (4) 30 JEE Main 2023 (29 Jan Shift 1) JEE Main Previous Year Paper

Q90.Let M be the maximum value of the product of two positive integers when their sum is 66 . Let the sample space S = {x ∈Z : x(66 −x) ≥59 M} and the event A = {x ∈S : x is a multiple of 3 }. Then P(A) is equal to (1) 15 (2) 1 44 3 (3) 1 (4) 7 5 22 JEE Main 2023 (25 Jan Shift 1) JEE Main Previous Year Paper